Graphs and Degree Equitability

Ahmad N. Al-Kenani, Nandappa D. Soner, Anwar Alwardi

Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.
Department of Studies in Mathematics, University of Mysore, Mysore, India.

DOI: 10.4236/am.2013.48160                

Abstract

Let G=(V,E)  be a graph. If φ is a function from the vertex set V(G) to the set of positive integers. Then two vertices u, v ∈ V(G)  are φ -equitable if｜φ(u)－φ(v)｜≤1.By the degree, equitable adjacency between vertices can be redefine almost all of the variants of the graphs. In this paper we study the degree equitability of the graph by defining equitable connectivity, equitable regularity, equitable connected graph and equitable complete graph. Some new families of graphs and some interesting results are obtained.

Keywords

Equitable Domination Number; Equitable Path; Equitable Walk; Equitable Connected Graph; EquitableRegular Graph; Equitable Complement Graph; Equitable Cut Vertex; Equitable Line Graph

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	[1] F. Harary, “Graph Theory,” Addison-Wesley, Reading, 1969.
[2]	[2] V. Swaminathan and K. M. Dharmalingam, “Degree Equitable Domination on Graphs,” Kragujevac Journal of Mathematics, Vol. 35, No. 1, 2011, pp. 191-197.